We study the ground state of a dilute Bose gas in a scaling limit where the Gross-Pitaevskii functional emerges. This is a repulsive non-linear Schrödinger functional whose quartic term is proportional to the scattering length of the interparticle interaction potential. We propose a new derivation of this limit problem, with a method that bypasses some of the technical difficulties that previous derivations had to face. The new method is based on a combination of Dyson's lemma, the quantum de Finetti theorem and a second moment estimate for ground states of the effective Dyson Hamiltonian. It applies equally well to the case where magnetic fields or rotation are present.We study the ground state of a dilute Bose gas in a scaling limit where the Gross-Pitaevskii functional emerges. This is a repulsive non-linear Schrödinger functional whose quartic term is proportional to the scattering length of the interparticle interaction potential. We propose a new derivation of this limit problem, with a method that bypasses some of the technical difficulties that previous derivations had to face. The new method is based on a combination of Dyson's lemma, the quantum de Finetti theorem and a second moment estimate for ground states of the effective Dyson Hamiltonian. It applies equally well to the case where magnetic fields or rotation are present.

Mar 31, 201565 citations

Bose-Einstein Condensation of Dilute Gases in Traps

Élliott H. Lieb, Robert Seiringer

The ground state of a gas of Bosons confined in an external trap potential and interacting via repulsive two-body forces has recently been shown to exhibit complete Bose-Einstein condensation in the dilute limit, yielding for the first time a rigorous proof of this phenomenon in a physically realistic setting. We give here an account of this work about the Gross-Pitaevskii limit where the particle number $N$ goes to infinity with $Na$ fixed, where $a$ is the scattering length of the interparticle interaction, measured in units of the trap size.The ground state of a gas of Bosons confined in an external trap potential and interacting via repulsive two-body forces has recently been shown to exhibit complete Bose-Einstein condensation in the dilute limit, yielding for the first time a rigorous proof of this phenomenon in a physically realistic setting. We give here an account of this work about the Gross-Pitaevskii limit where the particle number $N$ goes to infinity with $Na$ fixed, where $a$ is the scattering length of the interparticle interaction, measured in units of the trap size.

arXiv:math-ph/0210028Oct 13, 20020 citations

Ground state energy of a Bose gas in the Gross–Pitaevskii regime

Giulia Basti, Serena Cenatiempo, Alessandro Olgiati +2 more

We review some rigorous estimates for the ground state energy of dilute Bose gases. We start with Dyson’s upper bound, which provides the correct leading order asymptotics for hard spheres. Afterward, we discuss a rigorous version of Bogoliubov theory, which recently led to an estimate for the ground state energy in the Gross–Pitaevskii regime, valid up to second order, for particles interacting through integrable potentials. Finally, we explain how these ideas can be combined to establish a new upper bound, valid to second order, for the energy of hard spheres in the Gross–Pitaevskii limit. Here, we only sketch the main ideas; details will appear elsewhere.We review some rigorous estimates for the ground state energy of dilute Bose gases. We start with Dyson’s upper bound, which provides the correct leading order asymptotics for hard spheres. Afterward, we discuss a rigorous version of Bogoliubov theory, which recently led to an estimate for the ground state energy in the Gross–Pitaevskii regime, valid up to second order, for particles interacting through integrable potentials. Finally, we explain how these ideas can be combined to establish a new upper bound, valid to second order, for the energy of hard spheres in the Gross–Pitaevskii limit. Here, we only sketch the main ideas; details will appear elsewhere.

arXiv:2202.10270Journal of Mathematical PhysicsApr 1, 202211 citations

The Ground State of the Bose Gas

Élliott H. Lieb, Robert Seiringer, Jan Philip Solovej +1 more

Now that the low temperature properties of quantum-mechanical many-body systems (bosons) at low density, $ρ$ , can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4-5 decades ago. For systems with repulsive (i.e. positive) interaction potentials the experimental low temperature state and the ground state are effectively synonymous -- and this fact is used in all modeling. In such cases, the leading term in the energy/particle is $2π\hbar^2 a ρ/m$ where $a$ is the scattering length of the two-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange $N^{7/5}$ law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has recently been obtained by us and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two-dimensions and this, too, will be shown to be correct. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, two other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii equation correctly describes the ground state in the `traps' actually used in the experiments. For this system it is also possible to prove complete Bose condensation, as we have shown. Another topic is a proof that Foldy's 1961 theory of a high density Bose gas of charged particles correctly describes its ground state energy.Now that the low temperature properties of quantum-mechanical many-body systems (bosons) at low density, $ρ$, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4-5 decades ago. For systems with repulsive (i.e. positive) interaction potentials the experimental low temperature state and the ground state are effectively synonymous -- and this fact is used in all modeling. In such cases, the leading term in the energy/particle is $2π\hbar^2 a ρ/m$ where $a$ is the scattering length of the two-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange $N^{7/5}$ law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has recently been obtained by us and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two-dimensions and this, too, will be shown to be correct. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, two other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii equation correctly describes the ground state in the `traps' actually used in the experiments. For this system it is also possible to prove complete Bose condensation, as we have shown. Another topic is a proof that Foldy's 1961 theory of a high density Bose gas of charged particles correctly describes its ground state energy.

arXiv:math-ph/0204027Apr 11, 20025 citations

Gross-Pitaevskii non-linear dynamics for pseudo-spinor condensates

Alessandro Michelangeli, Alessandro Olgiati

We derive the equations for the non-linear effective dynamics of a so called pseudo-spinor Bose-Einstein condensate, which emerges from the linear many-body Schrödinger equation at the leading order in the number of particles.The considered system is a three-dimensional diluted gas of identical bosons with spin, possibly confined in space, and coupled with an external time-dependent magnetic field; particles also interact among themselves through a short-scale repulsive interaction.The limit of infinitely many particles is monitored in the physically relevant Gross-Pitaevskii scaling.In our main theorem, if at time zero the system is in a phase of complete condensation (at the level of the reduced one-body marginal) and with energy per particle fixed by the Gross-Pitaevskii functional, then such conditions persist also at later times, with the one-body orbital of the condensate evolving according to a system of non-linear cubic Schrödinger equations coupled among themselves through linear (Rabi) terms.The proof relies on an adaptation to the spinor setting of Pickl's projection counting method developed for the scalar case.Quantitative rates of convergence are available, but not made explicit because evidently non-optimal.In order to substantiate the formalism and the assumptions made in the main theorem, in an introductory section we review the mathematical formalisation of modern typical experiments with pseudo-spinor condensates.We derive the equations for the non-linear effective dynamics of a so called pseudo-spinor Bose-Einstein condensate, which emerges from the linear many-body Schrödinger equation at the leading order in the number of particles.The considered system is a three-dimensional diluted gas of identical bosons with spin, possibly confined in space, and coupled with an external time-dependent magnetic field; particles also interact among themselves through a short-scale repulsive interaction.The limit of infinitely many particles is monitored in the physically relevant Gross-Pitaevskii scaling.In our main theorem, if at time zero the system is in a phase of complete condensation (at the level of the reduced one-body marginal) and with energy per particle fixed by the Gross-Pitaevskii functional, then such conditions persist also at later times, with the one-body orbital of the condensate evolving according to a system of non-linear cubic Schrödinger equations coupled among themselves through linear (Rabi) terms.The proof relies on an adaptation to the spinor setting of Pickl's projection counting method developed for the scalar case.Quantitative rates of convergence are available, but not made explicit because evidently non-optimal.In order to substantiate the formalism and the assumptions made in the main theorem, in an introductory section we review the mathematical formalisation of modern typical experiments with pseudo-spinor condensates.

Journal of Nonlinear Mathematical PhysicsJun 28, 201718 citations

Gross-Pitaevskii non-linear dynamics for pseudo-spinor condensates

Alessandro Michelangeli, Alessandro Olgiati

We derive the equations for the non-linear effective dynamics of a so called pseudo-spinor Bose-Einstein condensate, which emerges from the linear many-body Schrödinger equation at the leading order in the number of particles. The considered system is a three-dimensional diluted gas of identical bosons with spin, possibly confined in space, and coupled with an external time-dependent magnetic field; particles also interact among themselves through a short-scale repulsive interaction. The limit of infinitely many particles is monitored in the physically relevant Gross-Pitaevskii scaling. In our main theorem, if at time zero the system is in a phase of complete condensation (at the level of the reduced one-body marginal) and with energy per particle fixed by the Gross-Pitaevskii functional, then such conditions persist also at later times, with the one-body orbital of the condensate evolving according to a system of non-linear cubic Schrödinger equations coupled among themselves through linear (Rabi) terms. The proof relies on an adaptation to the spinor setting of Pickl's projection counting method developed for the scalar case. Quantitative rates of convergence are available, but not made explicit because evidently non-optimal. In order to substantiate the formalism and the assumptions made in the main theorem, in an introductory section we review the mathematical formalisation of modern typical experiments with pseudo-spinor condensates.We derive the equations for the non-linear effective dynamics of a so called pseudo-spinor Bose-Einstein condensate, which emerges from the linear many-body Schrödinger equation at the leading order in the number of particles. The considered system is a three-dimensional diluted gas of identical bosons with spin, possibly confined in space, and coupled with an external time-dependent magnetic field; particles also interact among themselves through a short-scale repulsive interaction. The limit of infinitely many particles is monitored in the physically relevant Gross-Pitaevskii scaling. In our main theorem, if at time zero the system is in a phase of complete condensation (at the level of the reduced one-body marginal) and with energy per particle fixed by the Gross-Pitaevskii functional, then such conditions persist also at later times, with the one-body orbital of the condensate evolving according to a system of non-linear cubic Schrödinger equations coupled among themselves through linear (Rabi) terms. The proof relies on an adaptation to the spinor setting of Pickl's projection counting method developed for the scalar case. Quantitative rates of convergence are available, but not made explicit because evidently non-optimal. In order to substantiate the formalism and the assumptions made in the main theorem, in an introductory section we review the mathematical formalisation of modern typical experiments with pseudo-spinor condensates.

arXiv:1704.00150Apr 1, 20175 citations

N

goes to infinity with

Na

fixed, where

a

is the scattering length of the interparticle interaction, measured in units of the trap size.

Now that the low temperature properties of quantum-mechanical many-body systems (bosons) at low density,

ρ

, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4-5 decades ago. For systems with repulsive (i.e. positive) interaction potentials the experimental low temperature state and the ground state are effectively synonymous -- and this fact is used in all modeling. In such cases, the leading term in the energy/particle is

2π\hbar^2 a ρ/m

where

a

is the scattering length of the two-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange

N^{7/5}

law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has recently been obtained by us and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two-dimensions and this, too, will be shown to be correct. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, two other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii equation correctly describes the ground state in the `traps' actually used in the experiments. For this system it is also possible to prove complete Bose condensation, as we have shown. Another topic is a proof that Foldy's 1961 theory of a high density Bose gas of charged particles correctly describes its ground state energy.